Recently, I discussed attempts to stop explaining so much. Experiments with assessments are showing success. Skills are great, but I want to see who is thinking. Part of the reason I don't mind letting students (even PreAP, the horror!) use their notebooks on assessments is that a well done thinking question can't be anticipated. Through the course of working with the skills, the student either developed a strong working idea of the concept and therefore can answer just about anything, or they didn't.
Last week, variations of this question were posed:
Sure, you can slap that into y= and see. But it'll be obvious that's what you did when your explanation comes up short. Also, I never approached rationals that way. How do you interpret that ratio? What secrets will it give away?
There were three versions of this question. In two of them, the ratio straight up didn't work. The asymptote or discontinuity would show in the wrong place. This one though, will produce asymptotes at x = -7 and x = 3 as shown. 90% of students agreed and said, yes, this could be the correct graph because they factored the given ratio and saw the right looking numbers. Full credit was awarded. A small handful though...
While not explicitly being tested, we discussed a few days earlier how you get a general idea of a ratio's behavior by doing some testing on either side of the asymptote. Choose a test point on either side, and see if it would produce a positive or negative number. While not super accurate, it'll get you close.
It blew me away that these students saw something generally correct but still weren't satisfied. The behavior on either side needed to match too. While the top picture was inaccurate (-6 placed on the wrong side of -7), the student was suspicious, and I love it. The kid in the bottom picture nailed it exactly. I agree with the behavior for THIS asymptote, but hey man, this is the right number and all, but it's still no good. Disagree.
I mean, just wow.
Here's the bigger picture idea. It didn't occur to me to try this line of thinking with rationals until my 5th year teaching Pre-Cal (6th year overall, 4th year of listening to smart people on Twitter). When you say you want kids to really know the material, are you sure you've got a complete handle on it yourself?